ISSN:
1930-5311
eISSN:
1930-532X
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Journal of Modern Dynamics
October 2013 , Volume 7 , Issue 4
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2013, 7(4): 489-526
doi: 10.3934/jmd.2013.7.489
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Abstract:
Let ${\cal Q}$ be a connected component of a stratum in the moduli space of abelian or quadratic differentials for a nonexceptional Riemann surface $S$ of finite type. We prove that the probability measure on ${\cal Q}$ in the Lebesgue measure class which is invariant under the Teichmüller flow is obtained by Bowen's construction.
Let ${\cal Q}$ be a connected component of a stratum in the moduli space of abelian or quadratic differentials for a nonexceptional Riemann surface $S$ of finite type. We prove that the probability measure on ${\cal Q}$ in the Lebesgue measure class which is invariant under the Teichmüller flow is obtained by Bowen's construction.
2013, 7(4): 527-552
doi: 10.3934/jmd.2013.7.527
+[Abstract](1725)
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Abstract:
We analyze a class of $C^0$-small but $C^1$-large deformations of Anosov diffeomorphisms that break the topological conjugacy and structural stability, but unexpectedly retain the following stability property. The usual semiconjugacy mapping the deformation to the Anosov diffeomorphism is in fact an isomorphism with respect to all ergodic, invariant probability measures with entropy close to the maximum. In particular, the value of the topological entropy and the existence of a unique measure of maximal entropy are preserved. We also establish expansiveness around those measures. However, this expansivity is too weak to ensure the existence of symbolic extensions.
Many constructions of robustly transitive diffeomorphisms can be done within this class. In particular, we show that it includes a class described by Bonatti and Viana of robustly transitive diffeomorphisms that are not partially hyperbolic.
We analyze a class of $C^0$-small but $C^1$-large deformations of Anosov diffeomorphisms that break the topological conjugacy and structural stability, but unexpectedly retain the following stability property. The usual semiconjugacy mapping the deformation to the Anosov diffeomorphism is in fact an isomorphism with respect to all ergodic, invariant probability measures with entropy close to the maximum. In particular, the value of the topological entropy and the existence of a unique measure of maximal entropy are preserved. We also establish expansiveness around those measures. However, this expansivity is too weak to ensure the existence of symbolic extensions.
Many constructions of robustly transitive diffeomorphisms can be done within this class. In particular, we show that it includes a class described by Bonatti and Viana of robustly transitive diffeomorphisms that are not partially hyperbolic.
2013, 7(4): 553-563
doi: 10.3934/jmd.2013.7.553
+[Abstract](1398)
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Abstract:
Given any Liouville number $\alpha$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation-preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.
Given any Liouville number $\alpha$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation-preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.
2013, 7(4): 565-604
doi: 10.3934/jmd.2013.7.565
+[Abstract](1506)
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Abstract:
We consider a partially hyperbolic $C^1$-diffeomorphism $f\colon M \rightarrow M$ with a uniformly compact $f$-invariant center foliation $\mathcal{F}^c$. We show that if the unstable bundle is one-dimensional and oriented, then the holonomy of the center foliation vanishes everywhere, the quotient space $M/\mathcal{F}^c$ of the center foliation is a torus and $f$ induces a hyperbolic automorphism on it, in particular, $f$ is centrally transitive.
We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set of the quotient dynamics, and we establish the existence of a holonomy-invariant family of measures on the unstable leaves (Margulis measure).
We consider a partially hyperbolic $C^1$-diffeomorphism $f\colon M \rightarrow M$ with a uniformly compact $f$-invariant center foliation $\mathcal{F}^c$. We show that if the unstable bundle is one-dimensional and oriented, then the holonomy of the center foliation vanishes everywhere, the quotient space $M/\mathcal{F}^c$ of the center foliation is a torus and $f$ induces a hyperbolic automorphism on it, in particular, $f$ is centrally transitive.
We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set of the quotient dynamics, and we establish the existence of a holonomy-invariant family of measures on the unstable leaves (Margulis measure).
2013, 7(4): 605-618
doi: 10.3934/jmd.2013.7.605
+[Abstract](1503)
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Abstract:
We show that for every compact $3$-manifold $M$ there exists an open subset of $Diff^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures that are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher-regularity case, where Pesin Theory gives stable and unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity.
We show that for every compact $3$-manifold $M$ there exists an open subset of $Diff^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures that are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher-regularity case, where Pesin Theory gives stable and unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity.
2013, 7(4): 619-637
doi: 10.3934/jmd.2013.7.619
+[Abstract](1711)
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Abstract:
We consider cocycles $\tilde A: \mathbb{T}\times K^d \ni (x,v)\mapsto ( x+\omega, A(x,E)v)$ with $\omega$ Diophantine, $K=\mathbb{R}$ or $K=\mathbb{C}$. We assume that $A: \mathbb{T}\times \mathfrak{E} \to GL(d,K)$ is continuous, depends analytically on $x\in\mathbb{T}$ and is Hölder in $E\in \mathfrak{E} $, where $\mathfrak{E}$ is a compact metric space. It is shown that if all Lyapunov exponents are distinct at one point $E_{0}\in\mathfrak{E}$, then they remain distinct near $E$. Moreover, they depend in a Hölder fashion on $E\in B$ for any ball $B\subset \mathfrak{E}$ where they are distinct. Similar results, with a weaker modulus of continuity, hold for higher-dimensional tori $\mathbb{T}^\nu$ with a Diophantine shift. We also derive optimal statements about the rate of convergence of the finite-scale Lyapunov exponents to their infinite-scale counterparts. A key ingredient in our arguments is the Avalanche Principle, a deterministic statement about long finite products of invertible matrices, which goes back to work of Michael Goldstein and the author. We also discuss applications of our techniques to products of random matrices.
We consider cocycles $\tilde A: \mathbb{T}\times K^d \ni (x,v)\mapsto ( x+\omega, A(x,E)v)$ with $\omega$ Diophantine, $K=\mathbb{R}$ or $K=\mathbb{C}$. We assume that $A: \mathbb{T}\times \mathfrak{E} \to GL(d,K)$ is continuous, depends analytically on $x\in\mathbb{T}$ and is Hölder in $E\in \mathfrak{E} $, where $\mathfrak{E}$ is a compact metric space. It is shown that if all Lyapunov exponents are distinct at one point $E_{0}\in\mathfrak{E}$, then they remain distinct near $E$. Moreover, they depend in a Hölder fashion on $E\in B$ for any ball $B\subset \mathfrak{E}$ where they are distinct. Similar results, with a weaker modulus of continuity, hold for higher-dimensional tori $\mathbb{T}^\nu$ with a Diophantine shift. We also derive optimal statements about the rate of convergence of the finite-scale Lyapunov exponents to their infinite-scale counterparts. A key ingredient in our arguments is the Avalanche Principle, a deterministic statement about long finite products of invertible matrices, which goes back to work of Michael Goldstein and the author. We also discuss applications of our techniques to products of random matrices.
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